Pendulum Motion and Simple Harmonic Approximation

Technically the movement of a pendulum doesn’t quite match simple harmonic motion.

However, if the angle theta is small enough,  θ~sin(θ)

In fact for angles less than 15 degrees the error will be less than one percent.

Thus a simple pendulum essentially displays SHM (simple harmonic motion).

Such approximations are very useful in certain applications, especially in engineering, where such a small amount of error can be tolerable.

Fear and Physics

From mozzercork on flickr

(from mozzercork on flickr)

If you have a tough physics problem, you might hesitate to even start it.

That won’t help your cause though…..

Use lots of paper and just write down what comes to your mind.

It might be completely wrong, but it doesn’t matter, you’ll find out pretty quickly by doing simple checks (dimensional analysis, feasibility, etc)

Thomas Edison tried 10,000 materials for filaments before getting the electric light to work.

And complex physics problems can get somewhat ridiculous.

If you start a problem earlier, it will be in the back of your head.

Professor Barber even told us that he has gone to sleep and woken up with the answer.

Tension Free Body Diagram

If a vine is hung over a tree branch in the forest and two monkeys are hanging on both ends then the tension in the vine will be equal throughout the whole rope, since if it wasn’t the vine would snap or go slack.

The tension force for one monkey’s side will be equal and opposite to the tension force on the other monkey’s side.

The vine itself will also require a different amount of force to slide across the top of the branch depending on how the vine and the tree branch interact.

The force required for movement is unique depending on the materials (which defines a coefficient of friction).

Objects will have different coefficients of friction, denoted as mu, based on how they act together.

In analyzing situations it is good to draw a picture or “free body diagram” and decide what the axes are, including which directions should be understood as positive and negative.

The free body diagram should  only be applied to simple situations (generally one or a few objects involved).

When the situation is complicated, like the effect of huge numbers of electrons hitting an object, other theories come into play.

Even with fifty forces that are three dimensional, you would have to add up 150 components (x, y, z for each) and that wouldn’t be fun.

Simple Harmonic Motion SHO

Many things exhibit something that resembles simple harmonic motion (abbreviated SHO).

Waves rise and fall with a regular pattern and a slinky oscillates.

To analyze simple harmonic motion, we turn to the sine and cosine functions which are also periodic.

With a bit of tweaking, according to different amplitudes and starting points, a harmonic motion equation can describe such periodic functions.

However, such methods only work for small displacements.

When the displacements get large enough, nonlinear effects come into play.

Phil Kesten Physics 32 Class at Santa Clara University

Hi Neal,
Okay, the pressure’s on:  I told my students about your site!  And yes, I’m teaching 32 this quarter, so anything you add on SHM, fluids, gravitation, waves, light, sound…  they’ll be looking!

prk

In light of this news, I will start to talk about Physics 32 stuff.

As far as my connection to Dr. Kesten, he was my advisor at Santa Clara and I took physics 31, 32, 33, and 34 from him.

Right now I am doing an MS in applied physics.

If I am unclear, let me know and leave comments!

I can even make videos, but only if I feel it is worth my time for now.

Changing Coordinate Systems to Make Problems Easier

One day a physicist comes across a perplexed student in the library.

It seems that the student had only learned to place his axes with the positive y axis going up and the positive x axis going to the right.

Unfortunately this system wasn’t working so well with a situation involving pulleys and curved directions.

The physicist decides to help the engineer by telling him that the x-y coordinate system can be changed to better suit the problem at hand, much like polar coordinates can be more convenient than cartesian coordinates.

If an object travels in a curved path he reasons that the x-y coordinate system may also be curved in parts for simplicity.

Gravity may be thought of as positive or negative and up may be considered down.

The physicist also says that simply using common sense to check an answer will often times be effective, for instance if a block is sitting on a wall with no forces it will probably just stay there.

Newtons Second Law F=ma

As the village blacksmith walks along one day he sees a group of monks circling around an apple tree and chanting something about the truth being expanded.

When he stops and listens he learns that a man named Newton became inspired by falling of apples.

The monks explain that Newton then came up with the equation ΣF = ma (where F and a are both vectors) which means that the sum of the forces in a direction is equal to the mass, not to be confused with weight, multiplied by the acceleration, one example being gravity, in a direction.

The F and a are vectors, indicated by bold.

The monks are rejoicing since they had only previously been able to understand the motion of objects.

Now they can understand interaction of objects with mass through the use of vector components, the old testament of the truth, and their newfound equation.

Circular Motion Centripetal Acceleration Arc Length

One day a little kid is playing with a yo-yo and swinging it around in a circle for his own amusement.

Suddenly a crazed physicist walks up and decides to explain the physics of the circular motion.

First he asks the kid to spin the yo-yo around without any acceleration and the kids tries, but this turns out to be a trick question since there always is an acceleration towards the center (centripetal or center seeking force) because the velocity changes directions.

The physicist next shows how components can be used for measuring velocity vectors.

He then explains that describing certain aspects of circular motion with rectangular coordinates is pure silliness and that sometimes polar coordinates with an angle and a radius are much simpler.

He notes however that unlike in navigation, the angle is measured from the positive x-axis in physic.

Finally the physicist gives a nifty trick (s=rφ) for finding the arc length of pizza, or any circle, if one knows the radius and angle phi, but speaking of pizza he thinks about how good some would be and walks off.

THE TRUTH in 3D Velocity and Acceleration Equations

Morpheus, after running the “Instant Kesten Program,” approaches his student Neo with two pills, a red one and a blue one.

He states that if he swallows the blue pill he will forget what has happened and return to his life, but if Neo swallows the red pill then THE TRUTH will split into multiple dimensions.

Neo swallows the red pill and suddenly the world becomes a bit more complicated, THE TRUTH breaks apart into

vx=vox
vy=voy-gt

x-x0=voxt

y-y0=voyt-1/2gt2.

The red pill makes him understand that horizontal movement has no acceleration due to gravity (and generally in simpler problems), but that vertical motion is affected by gravity.

Neo realizes that two dimensions, x and y, can also be combined into the formula

y – y0=(x-x0)tan(θ) – [g(x-x0)2]/[2v02cos2(θ)] when the acceleration present is gravity.

Morpheus tells him as well that multidimensional physics necessitates proper labeling of variables like time in order to be accurate.

Neo says “whoa.”

Basketball “Hang Time” Physics

Michael Jordan Flies Through the Air (statue) from Esparta on Flickr

Michael Jordan Flies Through the Air (statue) from Esparta on Flickr

The stadium is packed with fans as Michael Jordan flies through the air on his way to the rim. Gravity seems suspended in this “hang time” phenomenon.

The hot dog dealer casually explains that by using THE TRUTH, and possibly a know/don’t know table, it can easily be calculated that it is true that Jordan spends 2/3 of the time actually in the air and with a vertical leap of over forty inches this can be a long time.

The hot dog dealer continues to explain the physics of the game by showing how vector analysis can track the motion of basketball stars.

A vector has direction and magnitude and can be added or subtracted using normal additive rules.

Thus he explains that the offense can jump both forward and up, but if the defense has the same vertical component they will both land at the same time since horizontal and vertical motion components are independent.

The dealer then continues on his way and warns not to use tangent for the x or y components

Instead, multiply the vector by sin(θ) to find the y component (magnitude in the y direction) and multiply the vector by cos(θ) for the  x component (magnitude in the x direction.

Also, by convention you should measure θ from the positive x-axis (this is physics convention anyway).

Michael Jordan top 40 moments